Group #15689862 — subgroups

Subgroups — #15689862

Order 3008.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094020	C₂	1	1	yes	maximal, characteristic, complemented
2	#13094017	C₂	1	2	yes	maximal, complemented
3	#13094006	C₂	1	2	yes	maximal, complemented
4	#13094005	C₂	1	1	yes	maximal, characteristic, complemented
5	#13093997	C₂	1	2	yes	maximal, complemented
6	#13093996	C₂	1	2	yes	maximal, complemented
7	#13093995	C₂	1	2	yes	maximal, complemented
8	#13093993	C₂	1	2	yes	maximal, complemented
9	#120×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₄₇	C₂²	1	2	yes	—
12	C₂×D₄×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₄₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₄₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₄₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊D₄₇	C₂²	1	2	yes	complemented
17	C₂²×Dic₄₇	C₂²	1	1	yes	characteristic, complemented
18	C₂×D₁₈₈	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₄×D₄₇	C₂²	1	2	yes	complemented
20	C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²⋊Dic₄₇	C₂²	1	2	yes	—
23	C₂²⋊Dic₄₇	C₂²	1	2	yes	—
24	C₂²⋊Dic₄₇	C₂²	1	2	yes	—
25	C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic, complemented
26	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	complemented
27	C₄⋊Dic₄₇	C₂²	1	2	yes	complemented
28	Dic₄₇⋊C₄	C₂²	1	2	yes	complemented
29	Dic₄₇⋊C₄	C₂²	1	2	yes	complemented
30	C₄×Dic₄₇	C₂²	1	1	yes	characteristic, complemented
31	C₂³×C₄₇	—	4	2	no	—
32	C₂³×C₄₇	—	4	1	no	—
33	C₂³×C₄₇	C₂³	1	2	yes	—
34	C₂³×C₄₇	C₂³	1	1	yes	characteristic
35	C₂³×C₄₇	C₂³	1	1	yes	characteristic
36	C₂²×D₄₇	C₂³	1	2	yes	—
37	D₄×C₄₇	—	4	2	no	—
38	D₄×C₄₇	—	4	1	no	—
39	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₄₇	C₂³	1	2	yes	—
41	C₂²⋊D₄₇	—	4	2	no	—
42	C₂²⋊D₄₇	—	4	2	no	—
43	C₂²⋊D₄₇	—	4	2	no	—
44	C₂²⋊D₄₇	—	4	2	no	—
45	C₂×Dic₄₇	—	4	1	no	—
46	C₂×Dic₄₇	C₂³	1	2	yes	—
47	C₂×Dic₄₇	C₂³	1	2	yes	—
48	C₂×Dic₄₇	C₂³	1	2	yes	—
49	D₁₈₈	—	4	1	no	—
50	C₄×D₄₇	—	4	2	no	—
51	C₂²×C₄₇	—	2	1	no	—
52	C₂²×C₄₇	—	2	2	no	—
53	C₂²×C₄₇	—	2	2	no	—
54	C₂²×C₄₇	—	2	2	no	—
55	C₂²×C₄₇	—	2	2	no	—
56	C₂²×C₄₇	—	2	2	no	—
57	C₂²×C₄₇	—	2	1	no	—
58	C₂²×C₄₇	—	4	4	no	—
59	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
60	C₂×D₄₇	—	2	2	no	—
61	C₂×D₄₇	—	2	2	no	—
62	C₂×D₄₇	—	2	2	no	—
63	C₄×C₄₇	—	2	2	no	—
64	C₄×C₄₇	—	2	1	no	—
65	Dic₄₇	—	2	2	no	—
66	Dic₄₇	—	2	2	no	—
67	Dic₄₇	—	2	2	no	—
68	C₂×C₄₇	—	4	1	no	—
69	C₂×C₄₇	—	4	2	no	—
70	C₂×C₄₇	—	4	1	no	—
71	C₂×C₄₇	H(5,𝔽₂)	1	2	yes	—
72	C₂×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
73	D₄₇	—	4	2	no	—
74	#560	—	47	1	no	maximal, Sylow
75	C₄₇	#560	1	1	yes	characteristic, complemented, Sylow
76	#127	—	47	1	no	—
77	#126	—	47	2	no	—
78	#123	—	47	1	no	—
79	#123	—	47	2	no	—
80	#121	—	47	2	no	—
81	#121	—	47	2	no	—
82	#121	—	47	2	no	—
83	#120	—	47	2	no	—
84	#120	D₄₇	1	1	yes	characteristic, complemented
85	C₂⁴	C₂×D₄₇	1	1	yes	characteristic
86	C₂×D₄	—	47	2	no	—
87	C₂×D₄	—	47	2	no	—
88	C₂×D₄	—	47	2	no	—
89	C₂×D₄	—	47	2	no	—
90	C₂×D₄	—	47	1	no	—
91	C₂×D₄	C₂×D₄₇	1	2	yes	—
92	C₂×D₄	C₂×D₄₇	1	1	yes	characteristic
93	C₂²×C₄	—	47	1	no	—
94	C₂²×C₄	—	47	2	no	—
95	C₄⋊C₄	—	47	2	no	—
96	C₄⋊C₄	—	47	2	no	—
97	C₄⋊C₄	—	47	2	no	—
98	C₂²⋊C₄	—	47	2	no	—
99	C₂²⋊C₄	—	47	2	no	—
100	C₂²⋊C₄	—	47	2	no	—
101	C₂²⋊C₄	—	47	1	no	—
102	C₂²⋊C₄	—	47	2	no	—
103	C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
104	C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
105	C₄²	—	47	1	no	—
106	C₂³	—	4	2	no	—
107	C₂³	—	4	1	no	—
108	C₂³	—	47	2	no	—
109	C₂³	C₂²×D₄₇	1	2	yes	—
110	C₂³	C₂²×D₄₇	1	1	yes	characteristic
111	C₂³	C₂²×D₄₇	1	1	yes	characteristic
112	D₄	—	4	2	no	—
113	D₄	—	4	1	no	—
114	D₄	—	188	1	no	—
115	D₄	—	188	2	no	—
116	D₄	—	188	2	no	—
117	D₄	—	188	2	no	—
118	D₄	—	188	2	no	—
119	C₂×C₄	—	47	2	no	—
120	C₂×C₄	—	47	2	no	—
121	C₂×C₄	—	47	2	no	—
122	C₂×C₄	—	188	2	no	—
123	C₂×C₄	—	188	1	no	—
124	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₄₇	1	2	yes	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	4	4	no	—
134	C₂²	—	94	2	no	—
135	C₂²	—	94	2	no	—
136	C₂²	—	94	2	no	—
137	C₂²	C₂³×D₄₇	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	94	2	no	—
141	C₄	—	94	2	no	—
142	C₄	—	94	2	no	—
143	C₂	—	4	2	no	—
144	C₂	—	4	1	no	—
145	C₂	—	4	1	no	—
146	C₂	—	188	2	no	—
147	C₂	#13094069	1	2	yes	—
148	C₂	#13094069	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13094020

C₂

yes

maximal, characteristic, complemented

#13094017

C₂

yes

maximal, complemented

#13094006

C₂

yes

maximal, complemented

#13094005

C₂

yes

maximal, characteristic, complemented

#13093997

C₂

yes

maximal, complemented

#13093996

C₂

yes

maximal, complemented

#13093995

C₂

yes

maximal, complemented

#13093993

C₂

yes

maximal, complemented

#120×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂⁴×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

C₂²

yes

—

C₂×D₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊D₄₇

C₂²

yes

—

C₂×C₂²⋊D₄₇

C₂²

yes

—

C₂×C₂²⋊D₄₇

C₂²

yes

—

C₂×C₂²⋊D₄₇

C₂²

yes

complemented

C₂²×Dic₄₇

C₂²

yes

characteristic, complemented

C₂×D₁₈₈

C₂²

yes

characteristic, complemented

C₂×C₄×D₄₇

C₂²

yes

complemented

C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂²⋊C₄×C₄₇

C₂²

yes

characteristic

C₂²⋊Dic₄₇

C₂²

yes

—

C₂²⋊Dic₄₇

C₂²

yes

—

C₂²⋊Dic₄₇

C₂²

yes

—

C₂²⋊Dic₄₇

C₂²

yes

characteristic, complemented

(C₂×D₄₇)⋊C₄

C₂²

yes

complemented

C₄⋊Dic₄₇

C₂²

yes

complemented

Dic₄₇⋊C₄

C₂²

yes

complemented

Dic₄₇⋊C₄

C₂²

yes

complemented

C₄×Dic₄₇

C₂²

yes

characteristic, complemented

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂³

yes

—

C₂³×C₄₇

C₂³

yes

characteristic

C₂³×C₄₇

C₂³

yes

characteristic

C₂²×D₄₇

C₂³

yes

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

—

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

D₁₈₈

—

C₄×D₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂⁴

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

H(5,𝔽₂)

yes

—

C₂×C₄₇

H(5,𝔽₂)

yes

characteristic

D₄₇

—

#560

—

maximal, Sylow

C₄₇

#560

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₂⁴

C₂×D₄₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₇

yes

—

C₂×D₄

C₂×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

C₂×D₄₇

yes

—

104

C₂²⋊C₄

C₂×D₄₇

yes

characteristic

105

C₄²

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

C₂²×D₄₇

yes

—

110

C₂³

C₂²×D₄₇

yes

characteristic

111

C₂³

C₂²×D₄₇

yes

characteristic

112

D₄

—

113

D₄

—

114

D₄

—

188

—

115

D₄

—

188

—

116

D₄

—

188

—

117

D₄

—

188

—

118

D₄

—

188

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

188

—

123

C₂×C₄

—

188

—

124

C₂×C₄

C₂²×D₄₇

yes

characteristic

125

C₂×C₄

C₂²×D₄₇

yes

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂³×D₄₇

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

—

188

—

147

C₂

#13094069

yes

—

148

C₂

#13094069

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	#120×C₄₇	C₂
hypercenter	#120	D₄₇
hypocenter	C₄₇	#560
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #120

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₇

commutator, socle, cosocle

C₂²×C₄₇

C₂⁴

Fitting

#120×C₄₇

C₂

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂² ◁ #120

N	H ≅ G/N	type
1. #13094020	C₂	semidirect
2. #13094017	C₂	semidirect
3. #13094006	C₂	semidirect
4. #13094005	C₂	semidirect
5. #13093997	C₂	semidirect
6. #13093996	C₂	semidirect
7. #13093995	C₂	semidirect
8. #13093993	C₂	semidirect
9. #120×C₄₇	C₂	semidirect
16. C₂×C₂²⋊D₄₇	C₂²	semidirect
17. C₂²×Dic₄₇	C₂²	semidirect
18. C₂×D₁₈₈	C₂²	semidirect
19. C₂×C₄×D₄₇	C₂²	semidirect
25. C₂²⋊Dic₄₇	C₂²	semidirect
26. (C₂×D₄₇)⋊C₄	C₂²	semidirect
27. C₄⋊Dic₄₇	C₂²	semidirect
28. Dic₄₇⋊C₄	C₂²	semidirect
29. Dic₄₇⋊C₄	C₂²	semidirect
30. C₄×Dic₄₇	C₂²	semidirect
75. C₄₇	#560	semidirect
84. #120	D₄₇	semidirect

H ≅ G/N

type

1. #13094020

C₂

semidirect

2. #13094017

C₂

semidirect

3. #13094006

C₂

semidirect

4. #13094005

C₂

semidirect

5. #13093997

C₂

semidirect

6. #13093996

C₂

semidirect

7. #13093995

C₂

semidirect

8. #13093993

C₂

semidirect

9. #120×C₄₇

C₂

semidirect

16. C₂×C₂²⋊D₄₇

C₂²

semidirect

17. C₂²×Dic₄₇

C₂²

semidirect

18. C₂×D₁₈₈

C₂²

semidirect

19. C₂×C₄×D₄₇

C₂²

semidirect

25. C₂²⋊Dic₄₇

C₂²

semidirect

26. (C₂×D₄₇)⋊C₄

C₂²

semidirect

27. C₄⋊Dic₄₇

C₂²

semidirect

28. Dic₄₇⋊C₄

C₂²

semidirect

29. Dic₄₇⋊C₄

C₂²

semidirect

30. C₄×Dic₄₇

C₂²

semidirect

75. C₄₇

#560

semidirect

84. #120

D₄₇

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13094020	2
#13094017	2
#13094006	2
#13094005	2
#13093997	2
#13093996	2
#13093995	2
#13093993	2
#120×C₄₇	2
#560	47

subgroup

index

#120×C₄₇

Subgroups — #15689862

Conjugacy classes of subgroups

Complements

Maximal subgroups